Turn geometry into arithmetic
Coordinates encode position. Coordinate differences form horizontal and vertical legs of a right triangle, allowing Pythagoras to calculate straight-line distance.
Learning goals
- Interpret axes, projections and coordinate distances.
- Derive and use the distance formula accurately.
- Solve unknown-coordinate and equidistance conditions.
- Use calculated distances to verify collinearity or a geometric figure.
Concept lesson
Ordered pairs and axes
In (x,y), x is horizontal and y vertical. On the x-axis y=0; on the y-axis x=0; the origin is (0,0).
Signed coordinate differences
Write each subtraction with brackets: 4-(-3)=7. Squaring removes the sign only after the difference has been calculated.
Projection and distance from axes
The perpendicular projection of (a,b) on the x-axis is (a,0), and its distance from that axis is |b|. Its projection on the y-axis is (0,b), and its distance from that axis is |a|.
Derive the distance formula
For P(x1,y1) and Q(x2,y2), the horizontal and vertical changes are x2-x1 and y2-y1. Pythagoras gives PQ=sqrt((x2-x1)^2+(y2-y1)^2). Reversing both differences gives the same distance.
Distance evidence for geometric conclusions
Collinearity can be shown when one pairwise distance equals the sum of the other two. Equal side distances support isosceles or equilateral conclusions. A square requires four equal sides plus a condition ensuring right angles, such as equal diagonals; state exactly what the calculated distances prove.
Distance across negative coordinates
Between (-5,7) and (-1,3), use x-change=-1-(-5)=4 and y-change=3-7=-4. Distance=sqrt(4^2+(-4)^2)=sqrt(32)=4sqrt(2).
Unknown coordinate from a given distance
Substitute the known distance and coordinates, square both sides, simplify to an equation in the unknown, and retain every real value satisfying the original distance. A coordinate may have two valid positions.
Verify a collinear or isosceles configuration
Calculate only the needed squared distances first. For collinearity, compare the largest distance with the sum of the other two after taking roots when necessary. For an isosceles triangle, identify the equal pair and confirm the three points do not collapse into a line.
Axes and projections
For points in all four quadrants, give projections on both axes and non-negative perpendicular distances from each axis.
Bracket every subtraction
Correct four sign-error solutions by writing x2-x1 and y2-y1 with substituted coordinates in brackets before squaring.
Match calculations to a sufficient conclusion
Given computed sides and diagonals for several point sets, decide which named figure is established, which is ruled out, and where more evidence is required.
Locus of points equidistant from two fixed points
Set the two distance expressions equal, square once, cancel matching x-squared and y-squared terms, and simplify to a linear relation. Interpret it as the perpendicular-bisector locus.
One-, two- and three-mark distance patterns
For one mark, use axis facts directly. For two marks, show formula, substitution and simplification. For three marks, organise multiple distances and connect them explicitly to the geometric property being verified.
Coordinates and distance mastery
Project a signed-coordinate point onto an axis, calculate a distance, solve one unknown-coordinate condition, and verify a geometric claim from three or four points with sufficient distance evidence.